5-4 Dividing Polynomials
Objectives
To divide polynomials using long division
To divide polynomials using synthetic division
Image Long Description
Long division is one of many methods you can use to divide whole numbers.
Essential Understanding You can divide polynomials using steps that are similar to the long-division steps that you use to divide whole numbers.
When you try to factor a polynomial, you are trying to find a divisor of the polynomial that gives a quotient (the other factor) and remainder 0. This suggests that being able to divide one polynomial by another could help you factor polynomials.
Numerical long division and polynomial long division are similar.
Numerical Long Division
21
32
672
63
_
42
42
_
0
21
divides into
67
3
times
21
divides into
42
2
times
Polynomial Long Division
2
x
+
1
3
x
+
2
6
x
2
+
7
x
+
2
6
x
2
+
3
x
_
4
x
+
2
4
x
+
2
_
0
(
2
x
+
1
)
divides into
(
6
x
2
+
7
x
)
3
x
times
(
2
x
+
1
)
divides into
(
4
x
+
2
)
2
times
table with 1 row and 2 columns , row1 column 1 , table with 2 rows and 1 column , row1 column 1 , cap numericalcap longcap division , row2 column 1 , table with 5 rows and 1 column , row1 column 1 , 21 . table with 2 rows and 1 column , row1 column 1 , 32 , row2 column 1 , long division symbol enclosing , 672 , end symbol , end table , row2 column 1 , 63 with under bar below , row3 column 1 , 42 , row4 column 1 , 42 with under bar below , row5 column 1 , 0 , end table , column 2 table with 4 rows and 1 column , row1 column 1 , 21 . dividesinto , row2 column 1 , 67 3 , times , row3 column 1 , 21 . dividesinto , row4 column 1 , 42 2 , times , end table , end table , column 2 table with 2 rows and 1 column , row1 column 1 , cap polynomialcap longcap division , row2 column 1 , table with 5 rows and 1 column , row1 column 1 , 2 x plus 1 . table with 2 rows and 1 column , row1 column 1 , 3 x plus 2 , row2 column 1 , long division symbol enclosing , 6 , x squared , plus 7 x plus 2 , end symbol , end table , row2 column 1 , modified 6 , x squared , plus 3 x with under bar below , row3 column 1 , 4 x plus 2 , row4 column 1 , modified 4 x plus 2 with under bar below , row5 column 1 , 0 , end table , column 2 table with 4 rows and 1 column , row1 column 1 , open , 2 x plus 1 , close . dividesinto , row2 column 1 , open . 6 , x squared , plus 7 x . close . 3 x , times , row3 column 1 , open , 2 x plus 1 , close . dividesinto , row4 column 1 , open , 4 x plus 2 , close . 2 , times , end table , end table , end table
The remainder from each division above is 0, so 21 is a factor of 672 and
2
x
+
1
2 x plus 1 is a factor of
6
x
2
+
7
x
+
2
.
6 x squared , plus 7 x plus 2 .
Dynamic Activity
Lesson Vocabulary